The Palindromic Trees
The family of trees with palindromic characteristic polynomials is characterized. Large families of graphs with this property are found as well.
math.CO↗
arXiv subjects
Eduardo A. Canale
Publications and source records attributed to Eduardo A. Canale.
Counterexample to a Boesch's Conjecture
A key issue in network reliability analysis. A graph with $n$ nodes and whose $e$ edges fail independently with probability $p$ is an \emph{Uniformly Most Reliable Graph} (UMRG) if it has the highest reliability among all graphs with the same order and size for every value of $p$. The \emph{all-terminal reliability} is a polynomial in $p$ which defines the probability of a network to remain connected if some of its components fail. If the coefficients of the reliability polynomial are maximized by a graph, that graph is called \textit{Strong Uniformly Most Reliable Graph} (SUMRG) and it should be UMRG. An exhaustive computer search of the SUMRG with vertices up to 9 is done. Regular graphs with 10 to 14 vertices that maximize tree number are proposed as candidates to UMRG. As an outstanding result a UMRG with 9 vertices and 18 edges which has girth 3 is found, so smaller than the conjectured by Boesch in 1986. A new conjecture about UMRG's topology is posed here: the $(n,e)$-UMRG is $\overline{(k-1)C_3\cup C_{3+r}}$ whenever $n=3k+r$,$n\geq5$ and $e={n(n-3)}/{2}$. A reformulation of Boesch's conjecture is presented stating that if a $(n, {kn}/{2})$-UMRG exists and it has girth $g$, then it has maximum girth among all $k$-regular $(n,{kn}/{2})$ graphs and minimum number of $g$-cycles among those $k$-regular $(n,{kn}/{2})$ graphs with girth $g$.
math.CO↗
From weighted to unweighted graphs in Synchronizing Graph Theory
A way to associate unweighted graphs from weighted ones is presented, such that linear stable equilibria of the Kuramoto homogeneous model associated to both graphs coincide, i.e., equilibria of the system $\dotθ_i = \sum_{j \sim i} \sin(θ_{j}-θ_j)$, where $i\sim j$ means vertices $i$ and $j$ are adjacent in the corresponding graph. As a consequence, the existence of linearly stable equilibrium is proved to be NP-Hard as conjectured by R. Taylor in 2015 and a new lower bound for the minimum degree that ensures synchronization is found.
math.CO↗
Tromino Tilings with Pegs via Flow Networks
A tromino tiling problem is a packing puzzle where we are given a region of connected lattice squares and we want to decide whether there exists a tiling of the region using trominoes with the shape of an L. In this work we study a slight variation of the tromino tiling problem where some positions of the region have pegs and each tromino comes with a hole that can only be placed on top of the pegs. We present a characterization of this tiling problem with pegs using flow networks and show that (i) there exists a linear-time parsimonious reduction to the maximum-flow problem, and (ii) counting the number of such tilings can be done in linear-time. The proofs of both results contain algorithms that can then be used to decide the tiling of a region with pegs in $O(n)$ time.
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